generic two degree of freedomlinearandfuzzy
Post on 04-Jun-2018
216 Views
Preview:
TRANSCRIPT
-
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
1/24
Journal of the Franklin Institute ] (]]]]) ]]]]]]
Generic two-degree-of-freedom linear and fuzzy
controllers for integral processes
Radu-Emil Precupa,, Stefan Preitla, Emil M. Petriub,Jozsef K. Tarc, Marius L. Tomescud, Claudiu Poznae
aDepartment of Automation and Applied Informatics, Politehnica University of Timisoara, Bd. V. Parvan 2,RO-300223 Timisoara, Romania
bSchool of Information Technology and Engineering, University of Ottawa, 800 King Edward, Ottawa,
ON, Canada K1N 6N5cInstitute of Intelligent Engineering Systems, Budapest Tech Polytechnical Institution, Be csi ut 96/B,
H-1034 Budapest, HungarydFaculty of Computer Science, Aurel Vlaicu University of Arad, Complex Universitar M,
Str. Elena Dragoi 2, RO-310330 Arad, RomaniaeDepartment of Product Design and Robotics, Transilvania University of Brasov, Bd. Eroilor 28,
RO-500036 Brasov, Romania
Received 13 June 2008; received in revised form 16 February 2009; accepted 2 March 2009
Abstract
This paper presents a new framework for the design of generic two-degree-of-freedom (2-DOF),
linear and fuzzy, controllers dedicated to a class of integral processes specific to servo systems. The
first part of the paper presents four 2-DOF linear PI controller structures that are designed using
the Extended Symmetrical Optimum method to ensure the desired overshoot and settling time. The
second part of the paper presents an original design method for 2-DOF TakagiSugeno PI-fuzzy
controllers based on the stability analysis theorem. Experimental results for the speed control of aservo system with variable load illustrate the performance of the new generic control structures.
r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Keywords: Fuzzy control; PI and PID control; Servo systems; Tracking; Tuning
ARTICLE IN PRESS
www.elsevier.com/locate/jfranklin
0016-0032/$32.00 r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfranklin.2009.03.006
Corresponding author. Tel.: +40 256 40 3229; fax: +40 256 40 3214.E-mail address: radu.precup@aut.upt.ro (R.-E. Precup).
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://www.elsevier.com/locate/jfranklinhttp://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006mailto:radu.precup@aut.upt.rohttp://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006mailto:radu.precup@aut.upt.rohttp://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://www.elsevier.com/locate/jfranklin -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
2/24
1. Introduction
Linear PI and PID controllers are used nowadays to control approximately 90% of
industrial processes worldwide [1]. However, the performance of these PI and PID
controllers depends not only on the tuning parameters but also on additionalfunctionalities such as anti-windup, feedforward action and set-point filtering [2].
Two-degree-of-freedom (2-DOF) PI and PID controllers have an advantage over the
1-DOF ones from the point of view of achieving high performance in set-point tracking
and the regulation in the presence of disturbance inputs[35]. But, the main drawback of
2-DOF controllers is that the overshoot reduction is paid by a slower set-point response.
Fuzzy control provides cost-effective nonlinear control solutions to a large number of
industrial applications. It has the following advantages [69]:
the use of linguistic rules and approximate reasoning is a relatively simple and easy way
to solve concrete control problems,
there is no need for sophisticated models and control design tools, it is sometimes the only way to initially approach the control of a complex, uncertain or
even not well defined process.
The introduction of dynamics in the fuzzy controller (FC) structures in the form of PD-,
PI- or PID-fuzzy controllers allows to further improve the control system (CS)
performance [1014]. They are based on the Mamdani model for PD-FCs, PI-FCs and
PID-FCs. The TakagiSugeno model is also used for the design of PD-FCs, PI-FCs and
PID-FCs.There are two main approaches to the design of the PD-FCs, PI-FCs and PID-FCs:
the first one is based on the fact that in some well-stated conditions the approximateequivalence between linear and fuzzy controllers is generally acknowledged[15,16],
the second one relies on the consideration of these FCs as nonlinear PD, PI or PIDcontrollers with variable gains [17,18].
We are using the first approach, for two reasons: (i) the design methods for FCs based
on the merge between the knowledge on conventional linear PI controllers and the
experience of experts in controlling the processes are widely accepted, and (ii) bylinearization around some steady-state operating points, certain classes of processes can be
considered as linear ones with variable parameters. The parameter variance makes the
control of such integral processes a difficult but challenging task and the fuzzy control
represents an attractive solution in this context. This situation is not encountered in case of
other conventional control solutions that give satisfaction only under particular operating
regimes.
We are using the TakagiSugeno FC model [19]because it allows to develop low-cost
automation solutions. The low-cost aim concerns the linear dependence of each rule on the
inputs making them behave as bumpless interpolators between linear controllers [20].
The adopted 2-DOF fuzzy controllers have the following attractive advantages [2125]:
improve the CS performance with respect to the modifications of set-point and loaddisturbance inputs ensured by the FCs,
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]2
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
3/24
overcome of the mentioned drawback of 2-DOF controllers to ensure small overshootand settling time.
The new 2-DOF TakagiSugeno PI-FCs presented in this paper provide efficient low-cost
control solutions and offer transparency in controller design and tuning. These controllersare intended to a representative class of integral processes specific to servo systems. The
design method is based on the fuzzification of the 2-DOF linear PI controllers tuned in
terms of a 2-DOF formulation of the Extended Symmetrical Optimum (ESO) method[26].
The proposed new design method for these PI-FCs ensures a systematic design
framework due to
the controller structure, characterized by small number of membership functions andrules,
the design method itself, expressed in terms of relatively simple design steps.The proposed design method has a generality advantage over other TakagiSugeno FC
structures that usually employ combinations ofHN
and FC techniques[2732]to meet the
desired CS performance specifications. The transparency of the proposed design method is
essentially due to the fact that, unlike other fuzzy CS structures[20], it offers an easy to use
connection between the controller tuning parameters and the CS performance indices.
The paper is organized as follows. Section 2 presents the 2-DOF PI CS structures in
relation with the linear control case in terms of the ESO method. Section 3 presents four
2-DOF TakagiSugeno PI-FCs together with their design method highlighting the design
steps that correspond to the design method for 2-DOF linear PI controllers. The design isenabled by an original stability analysis theorem based on Lyapunovs theorem for time-
varying systems referred in [33,34]. Section 4 illustrates the case study dedicated to speed
control of a servo system with variable load. The proposed controllers and design methods
are validated by the inclusion of experimental results for low speed patterns and controllers
implemented as low-cost automation solutions. Section 5 discusses the conclusions.
2. Generic linear control system structures and extended symmetrical optimum method
The class of integral processes considered here is characterized by the transfer function P(s):
Ps kP=s1TSs, (1)wherekPis the controlled process gain and TSis the small time constant or the time constant
corresponding to the sum of parasitic time constants.
This class of integral processes characterizes servo systems encountered in many
practical applications including the speed control systems of hydro power generators,
control systems for electrical drives, position and speed control of mechatronics systems,
mobile robot guidance and control, etc. [3542]. Due to the parameter variance, the
control of this class of systems (1) is a challenging task when very good CS performance
indices are required. One solution to tackle this is the use of PI or PD control algorithms.
Acceptable CS performance indices with respect to the set-point and disturbance inputcan be obtained if the process with the transfer function P(s) is included to the generic
2-DOF PI CS structures presented in Fig. 1, referred to as the set-point filter structure
(Fig. 1(a)), the feedforward structure (Fig. 1(b)), the feedback structure (Fig. 1(c)) and the
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 3
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
4/24
component-separated structure (Fig. 1(d)). The variables in Fig. 1 are: r set-point,
y controlled output,eryorer1y control error,u control signal,r1,u2,u3,u4outputs of blocks F(s), C(s) in Fig. 1(b), C*(s) and CI(s), respectively, d1, d2, d3 load
disturbance input scenarios defined according toFig. 1(e), where the two blocks stand for
P1s kP1=1TSs; P2s kP2=s; kPkP1kP2. (2)The following nomenclature has been used for the blocks in all four 2-DOF PI CS
structures considered:
C(s) is the transfer function of the PI controller in Fig. 1(a) and (b):Cs kC11=Tis, (3)
with kC controller gain and Ti integral time constant, F(s) is the transfer function of the set-point filter in Fig. 1(a):Fs 1 1aTis=1Tis, (4)
with the design parameter a, andar1 to avoid the non-minimum phase character of
the closed-loop CS,
the other transfer functions in Fig. 1(b)(d):CFs kCa; Cs kC1a1=Tis; CPs 1a; CIs 1=Tis; CSs kC.
(5)
These four 2-DOF PI CS structures are equivalent schemes because they ensure the same
overall CS transfer functions, Gy,r(s) with respect to the set-point, Gy,d1(s) with respect to
d1, Gy,d2(s) with respect to d2 andGy,d3(s) with respect to d3:
Gy;r
s
1
a
Tis
1
=m
s
,
Gy;d1s Ti=kCs=ms; Gy;d2s kP2Ti=kC=kPsTSs1=ms,Gy;d3s Ti=kC=kPs2TSs1=ms; ms a3s3 a2s2 a1sa0,a01; a1Ti; a2Ti=kC=kP; a3TiTS=kC=kP. (6)
ARTICLE IN PRESS
Fig. 1. Set-point filter 2-DOF PI control system structure (a), feedforward 2-DOF PI control system structure
(b), feedback 2-DOF PI control system structure (c), component-separated 2-DOF PI control system structure
(d), definition of load disturbance input scenarios (e).
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]4
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
5/24
The open-loop transfer function L(s) takes the expression (7) in case of all four 2-DOF
PI CS structures:
Ls CsPs kCkPTis1=s2TiTSs1. (7)For the given class of processes with the transfer function in (1), making use of PI
controllers with the transfer function C(s) tuned in terms of Kesslers Symmetrical
Optimum (SO) method can ensure acceptable CS performance indices [2]. Kesslers SO
conditions concerning the denominator coefficients in (6):
2a0a2a21; 2a1a3a22 (8)were generalized to the following form specific to the ESO method [26]:
ffiffiffib
p a0a2a21;
ffiffiffib
p a1a3a22, (9)
where b represents a design parameter. The PI tuning conditions can be expressed as
kC 1=ffiffiffib
p kPTS; TibTS. (10)
Eq. (9) results in the following optimal expressions of the open-loop and closed-loop
transfer functions:
Ls 1bTSs=bffiffiffib
p T2Ss
21TSs; Gy;rs bTS1as1=mos,Gy;d1s b
ffiffiffib
p kPT
2Ss=mos; Gy;d2s kP2b
ffiffiffib
p T2SsTSs1=mos,
Gy;d3s bffiffiffib
p T2Ss
2TSs1=mos; mos b ffiffiffib
p T3Ss
3 bffiffiffib
p T2Ss
2 bTSs1.
(11)It is fully justified to consider the expressions (11) as optimal ones and the conditions (9)
as optimization ones because the conditions (9) ensure the maximization of the phase
margin in case of kPconst. The symmetry of open-loop Bode plots enables thisapproach; besides it guarantees a minimum desired phase margin in case of variable kP.
The definitions of overshoot and settling time depend on the CS inputs involved. They
are expressed using the general notations sv1 for the percent overshoot, sv1 for the
normalized overshoot, tvs for the normalized settling time, tvs for its normalized value,
where the superscript v stands for the current dynamic regime corresponding to the unit
step modification of the set-point, v
r, and disturbance inputs, v
d1, v
d2 orv
d3:
definitions with respect to the unit step modification of the set-point r:sr1sr1 100irxs1r; irx 1arexpbrx expdrxcr sinerx
fr coserx grexphrx; trsTStrs ; trs minfxtsrj jirx 1j 0:028 xxtsrg,(12)
wherexs1r is the solution to (13):
frer crdrsinerx crer frdrcoserx arbrexpdr brxg
r
hr
expdr
hr
x 0 (13)and the parameters in (12) and (13) have the expressions presented in Table 1with
Db2ffiffiffib
p 340, (14)
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 5
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
6/24
definitions with respect to the unit step modification of the disturbance input d1:sd11 kPTSsd11 ; sd11 100id1xs1d1; id1x ad1expbd1x expdd1xcd1 sined1x
fd1 cosed1x gd1 exphd1x; td1s TStd1s ; td1s minfxtsd1j jid1xj 0:028 xxtsd1g,(15)
where xs1d1 is the solution to (16):
fd1ed1 cd1dd1sined1x cd1ed1 fd1dd1cosed1x ad1bd1 expdd1 bd1xgd1hd1expdd1 hd1x 0 (16)
and the parameters in (15) and (16) have the expressions illustrated in Table 2withD
defined in (14),
definitions with respect to the unit step modification of the disturbance input d2:sd21 kP2TSsd21 ; sd21 100id2xs1d2; id2x ad2expb
d2
x expdd2
xcd2 sined2xfd2 cosed2x gd2exphd2x; td2s TStd2s ; td2s minfxtsd2j jid2xj 0:02 8 xxtsd2g,
(17)
ARTICLE IN PRESS
Table 1
Parameters in (12) and (13).
Parameter bo9 b9 b49
ar
ffiffiffibp a ffiffiffibp 1=3 ffiffiffibp 1 ffiffiffibp a ffiffiffibp 1=3 ffiffiffibp br 1=
ffiffiffib
p 1/3 1=
ffiffiffib
pcr a
ffiffiffib
p =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b2
ffiffiffib
pq 0 0dr 0:511=
ffiffiffib
p 1/3 0:5
ffiffiffib
p 1 ffiffiffiffiDp = ffiffiffibp
er0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b2
ffiffiffib
pq =
ffiffiffib
p 0 0fr 2a
ffiffiffib
p=3
ffiffiffib
p x/3 2 a1b
ffiffiffib
p
ffiffiffib
p ffiffiffiffiD
p =ffiffiffiffiD
p =b3
ffiffiffib
p
ffiffiffib
p ffiffiffiffiD
p 2
ffiffiffiffiD
p
gr 0 3a2x2=18 2 a1bffiffiffib
p
ffiffiffib
p ffiffiffiffiD
p =
ffiffiffiffiD
p =b3
ffiffiffib
p
ffiffiffib
p ffiffiffiffiD
p 2
ffiffiffiffiD
p
hr 0 1/3 0:5ffiffiffib
p 1
ffiffiffiffiD
p =ffiffiffib
p
Table 2
Parameters in (15) and (16)
Parameter bo9 b9 b49
ad1 b=3ffiffiffib
p 0.5x2 b=3
ffiffiffib
p
bd1 1=ffiffiffib
p 1/3 1=
ffiffiffib
pcd1
b=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 b2ffiffiffib
pq 0 0dd1 0:511=
ffiffiffib
p 0 0:5
ffiffiffib
p 1 ffiffiffiffiDp = ffiffiffibp
ed10:5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b2 ffiffiffibpq = ffiffiffibp
0 0
fd1 b=3ffiffiffib
p 0 2b= ffiffiffiffiDp = ffiffiffiffiDp 3 ffiffiffibp
gd1 0 0 2b=ffiffiffiffiD
p = ffiffiffiffiDp 3 ffiffiffibp
hd1 0 0 0:5ffiffiffib
p 1 ffiffiffiffiDp = ffiffiffibp
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]6
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
7/24
wherexs1d2 is the solution to
fd2ed2 cd2dd2sined2x cd2ed2 fd2dd2cosed2x ad2bd2expdd2 bd2x
gd2hd2exp
dd2
hd2
x
0 (18)
and the parameters in (17) and (18) have the expressions summarized in Table 3,
definitions with respect to the unit step modification of the disturbance input d3:sd31 sd31 100id3xs1d3; id3x ad3expbd3x expdd3xcd3 sined3x
fd3 cosed3x gd3exphd3x; td3s TStd3s ; td3s minfxtsd3j jid3xj 0:028 xxtsd3g,(19)
wherexs1d3 is the solution to
fd3ed3 cd3dd3sined3x cd3ed3 fd3dd3cosed3x ad3bd3expdd3 bd3xgd3hd3expdd3 hd3x 0 (20)
and the parameters in (19) and (20) have the expressions presented in Table 4.
ARTICLE IN PRESS
Table 3
Parameters in (17) and (18)
Parameter bo9 b9 b49
ad2
b
ffiffiffibp =3 ffiffiffibp x/3 b ffiffiffibp =3 ffiffiffibp bd2 1= ffiffiffibp 1/3 1= ffiffiffibpcd2 ffiffiffi
bp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3b 2ffiffiffib
pq =3
ffiffiffib
p 0 0
dd2 0:511=ffiffiffib
p 1/3 0:5
ffiffiffib
p 1 ffiffiffiffiDp = ffiffiffibp
ed20:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b2
ffiffiffib
pq =
ffiffiffib
p 0 0fd2 b
ffiffiffib
p =3
ffiffiffib
p x2/3
ffiffiffib
p ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp ffiffiffibp 3
gd2 0 0 ffiffiffi
bp
ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp 3 ffiffiffibp hd2 0 0 0:5
ffiffiffib
p 1 ffiffiffiffiDp = ffiffiffibp
Table 4
Parameters in (19) and (20)
Parameter bo9 b9 b49
ad3 1ffiffiffib
p =3
ffiffiffib
p x/3 1
ffiffiffib
p =3
ffiffiffib
p
bd3 1=ffiffiffib
p 1/3 1=
ffiffiffib
pcd3 2=3
ffiffiffib
p 0 0
dd3 0:511=ffiffiffib
p 1/3 0:5
ffiffiffib
p 1 ffiffiffiffiDp = ffiffiffibp
ed30:5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b 2 ffiffiffibpq = ffiffiffibp
0 0
fd3 0 x2/9 ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp ffiffiffibp 3gd3 0 1 ffiffiffiffiDp ffiffiffibp 1= ffiffiffiffiDp = ffiffiffiffiDp 3 ffiffiffibp hd3 0 1/3 0:5
ffiffiffib
p 1 ffiffiffiffiDp = ffiffiffibp
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 7
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
8/24
No simultaneous action of the disturbance inputs is supposed as far the expressions
(12)(20) are concerned. In addition, solving (13), (16), (18) and (20) requires numerical
techniques. Several diagrams for the normalized percent overshoot sv1 and normalizedsettling time tvs versus the design parametersa and b have been presented in[24].
The controller tuning of the 2-DOF PI CS structures presented inFig. 1is based on onlytwo design parameters, a and b, when the ESO method is applied.
Eqs. (12)(20) and Tables 14, together with a choice of design parameters a and b
within the domains ar1 and 1obo20, allow modifying the CS performance indices
represented by overshoot and settling time in a transparent manner according to designers
preferences. Since both the desired overshoot and the desired settling time should be
fulfilled, a trade-off to those performance indices can be defined.
To ensure the desired CS performance with respect to both the set-point r and one of the
disturbance inputs (d1,d2ord3) two classes of linear 2-DOF PI controllers can be designed:
the class of 2-DOF PI controllers, referred to as 2-DOF PI-C-r, tuned to ensure thedesired CS behaviour with respect to set-point in terms of the choice of design
parametersa and br and applying the tuning conditions obtained from (10) for bbr:
kC krC 1ffiffiffiffiffibr
p kPTS
; TiTri brTS, (21)
the class of 2-DOF PI controllers, referred to as 2-DOF PI-C-d, tuned to ensure thedesired CS behaviour with respect to disturbance inputs in terms of the choice of design
parametersa and bd and applying the tuning conditions obtained from (10) for b
bd:
kC kdC 1ffiffiffiffiffibd
q kPTS
; TiTdibdTS. (22)
Connecting the linear approach to fuzzy control, as an alternative control solution, the
2-DOF TakagiSugeno PI-FCs and a design method for these controllers will be proposed
in the following section. The 2-DOF TakagiSugeno PI-FCs belong to the class of type III
fuzzy systems[43,44]and blend the linear 2-DOF PI controllers separately designed with
respect to set-point and one of the disturbance inputs.
3. Generic fuzzy controller structures and design method
The 2-DOF TakagiSugeno PI-FCs play the roles of 2-DOF PI controllers in the CS
structures shown in Fig. 1 to improve the CS performance indices. These PI-FCs are
designed starting with the two classes of continuous-time 2-DOF PI controllers designed in
the previous section, 2-DOF PI-C-r and 2-DOF PI-C-d. Next, the dynamics components in
these linear 2-DOF PI controller structures are discretized as follows resulting in two
classes of quasi-continuous digital 2-DOF PI controllers with digital integral (I) and
proportional-integral (PI) components characterized by
the discrete-time equations of the digital PI components to substitute the blocks C(s)and C*(s) inFig. 1(a)(c) in case of 2-DOF PI-C-r:
Duk Durk Du2;k Dur2;k Du3;k Dur3;kKrPDekKrIek, (23)
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]8
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
9/24
the discrete-time equations of the digital PI components to substitute the blocks C(s)and C*(s) inFig. 1(a)(c) in case of 2-DOF PI-C-d:
Duk Dudk Du2;k Dud2;k Du3;k Dud3;kKdPDekKdIek, (24)
where Dek, Duk, Du2,k and Du3,k stand for the change of control error, of control
signal, ofu2 and u3, respectively:
Dek ekek1=h; Duk ukuk1=h; Du2;k u2;ku2:k1=h; Du3;k u3;ku3;k1=h,(25)
h is the sampling period, k is the index of current sampling interval, and the
parameters of the two incremental digital PI controllers (23) and (24) can be calculated
in terms of (26) in case of Tustins method and expressed in a unified manner for the
blocksC(s) and C*(s):
KrPkrC1ah=2Tri; KrI krC=Tri; KdPkdC1ah=2Tdi; KdI kdC=Tdi, (26)with a0 in case of blocks C(s) inFig. 1(a) and (b),
the discrete-time equations of the digital component to substitute the block F(s) inFig. 1(a):
r1;kg1r1;k1d0rkd1rk1, (27)where the parameters can be expressed by Tustins method:
d0 2Tri1a h=2Trih; d1 2Tri1a h=2Trih; g1 2Trih=2Trih,(28)
the discrete-time equations of the digital I components to substitute the block CI(s) inFig. 1(d) in case of 2-DOF PI-C-r:
Du4;k Dur4;k KrPDekKrIek, (29)
the discrete-time equations of the digital I components to substitute the block CI(s) inFig. 1(d) in case of 2-DOF PI-C-d:
Du4;k Dud4;k KdPDekKdIek, (30)
with the parameters
KrPh=2Tri; KrI 1=Tri; KdPh=2Tdi; KdI1=Tdi. (31)
The two classes of I and PI components are fuzzified resulting in four 2-DOF
TakagiSugeno PI-FC structures. These generic fuzzy controller structures are referred to
as: set-point filter 2-DOF PI-FC presented inFig. 2(a) (the fuzziffied version of the linear
controller structure in Fig. 1(a)), feedforward 2-DOF PI-FC illustrated in Fig. 2(b) (the
fuzziffied version of the linear controller structure in Fig. 1(b)), feedback 2-DOF PI-FC
shown inFig. 2(c) (the fuzziffied version of the linear controller structure inFig. 1(c)), and
component-separated 2-DOF PI-FC presented in Fig. 2(d) (the fuzziffied version of thelinear controller structure in Fig. 1(d)).
The key element inFig. 2is the basic four inputs two outputs fuzzy controller (B-FC)
that represents a TakagiSugeno fuzzy system. It makes use of the MAX and MIN
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 9
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
10/24
operators in the inference engine and it employs the weighted sum method for
defuzzification [4547]. The fuzzification is done in terms of the membership functions
illustrated in Fig. 3, where Drk is the change of set-point. Fig. 3 points out the
(positive) tuning parameters of the 2-DOF TakagiSugeno PI-FCs to be determined
by the design method, Se, SDe, SDr, Ss, and contributes to the fulfilment of the low-cost
aim.
Besides the task to elaborate Duk, Du2,k, Du3,kand Du4,k, the block B-FC has the task toobserve the current dynamic regimes of the CS. B-FC calculates the variable sk, with the
linguistic terms ZE and P corresponding to the dynamic regimes caused by the
modification of the disturbance inputs (of type d1, d2 or d3) and set-point r, respectively.
ARTICLE IN PRESS
Fig. 2. Set-point filter 2-DOF PI-fuzzy controller structure (a), feedforward 2-DOF PI-fuzzy controller structure (b),
feedback 2-DOF PI-fuzzy controller structure (c), component-separated 2-DOF PI-fuzzy controller structure (d).
Fig. 3. Input membership functions.
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]10
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
11/24
The variableskcan take two values, skSs40 when modifications ofr occur, andsk0when modifications ofd1, d2 or d3 occur. The variable sk is zero in two situations:
either the system is in steady-state regime and the set-point is constant,
or the set-point is constant and one of the disturbance inputs is variable.In all other situations skSs. This definition of the variable sk allows the simple
characterization of the dynamic regimes mentioned before. Use is made of the definition to
formulate the rule base of B-FC expressed in terms of the decision tables presented in
Tables 5 and 6.
The 2-DOF PI-FCs combine four linear 2-DOF PI controllers separately designed with
respect to the CS inputs, two of them which belong to the class 2-DOF PI-C-r and have the
design parameters b1 and b2, and the other two of them which belong to the class 2-DOF
PI-C-d and have the design parameters b3
andb4
. With this regard the rule consequentsl1
ARTICLE IN PRESS
Table 5
Decision table to calculate Duk, Du2,k, Du3,kand Du4,k.
|Drk|
ZE P
ek ek
N ZE P N ZE P
sk1P Dek P l1 l1 l2 l1 l1 l2ZE l1 l3 l1 l1 l1 l1N l2 l1 l1 l2 l1 l1
ZE Dek P l3 l3 l4 l1 l1 l2ZE l3 l3 l3 l1 l1 l1N l4 l3 l3 l2 l1 l1
Table 6
Decision table to calculate sk.
|Drk|
ZE P
ek ek
N ZE P N ZE P
sk1P Dek P Ss Ss Ss Ss Ss Ss
ZE Ss 0 Ss Ss Ss SsN S
s S
s S
s S
s S
s S
s
ZE Dek P 0 0 0 Ss Ss SsZE 0 0 0 Ss Ss SsN 0 0 0 Ss Ss Ss
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 11
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
12/24
andl2 correspond to 2-DOF PI-C-r, and l3 andl4 correspond to 2-DOF PI-C-d:
l1 Durk Dur2;k Dur3;k Dur4;kKrP1DekKrI1ek,l2
Durk
Dur2;k
Dur3;k
Dur4;k
KrP2Dek
KrI2ek,
l3 Dudk Dud2;k Dud3;k Dud4;kKdP1DekKdI1ek,l4 Dudk Dud2;k Dud3;k Dud4;kKdP2DekKdI2ek, (32)
where the additional subscript (1 and 2) inserted into the digital PI controller parameters
highlights a certain rule consequent.
The rule base to calculate Duk, Du2,k, Du3,kand Du4,kmaking use of the rule consequents
l1, l2, l3 and l4 in Table 5 can be interpreted in terms of four rules for each PI-FC
structure,Ru1, Ru2, Ru3 andRu4:
Ru1 :IF
ek IS N AND Dek IS P AND
jDrk
j IS ZE AND sk
1 IS P
OR
ek IS ZE AND Dek IS P AND jDrkj IS ZE AND sk1 IS P OR . . . ORek IS P AND Dek IS N AND jDrkj IS P AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl1,
Ru2 :IF ek IS P AND Dek IS P AND jDrkj IS ZE AND sk1 IS P ORek IS P AND Dek IS P AND jDrkj IS P AND sk1 IS P OR . . . ORek IS N AND Dek IS N AND jDrkj IS P AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl2,
Ru3 :IF
ek
IS ZE AND Dek
IS ZE ANDjDr
kj IS ZE AND s
k1 IS P
OR
ek IS N AND Dek IS P AND jDrkj IS ZE AND sk1 IS P OR . . . ORek IS P AND Dek IS N AND jDrkj IS ZE AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl3,
Ru4 :IF ek IS P AND Dek IS P AND jDrkj IS ZE AND sk1 IS ZE ORek IS N AND Dek IS N AND jDrkj IS ZE AND sk1 IS ZE THENDuk Du2;k Du3;k Du4;kl4, (33)
with the parameters in the rule consequents calculated according to (21), (22) and (26)
under the conditionsbr b1 in case of l1; br b2 in case of l2; bd b3 in case of l3; bd b4 in case of l4.
(34)
The rule base to calculate the variable skcan be interpreted as the following two rules
valid for each PI-FC structure:
Rs1 :IF ek IS N AND Dek IS P AND jDrkj IS ZE AND sk1 IS P ORek IS ZE ANDDek IS P AND jDrkj IS ZE AND sk1 IS P OR . . . ORek IS P AND Dek ISN AND
jDrk
j IS P AND sk1 IS ZE
THEN sk
Ss
Rs2 :IF ek IS ZE AND Dek IS ZE ANDjDrkj IS ZE AND sk1 IS P ORek IS NAND Dek IS P AND jDrkj IS ZE AND sk1 IS ZE OR . . . ORek IS P ANDDek IS N AND jDrkj IS ZE AND sk1 IS ZE THEN sk0. (35)
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]12
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
13/24
The presence of additional rules with the control laws l2 andl4 in their consequents is
necessary in order to alleviate the overshoot that appears when ekand Dektake the same
sign. This aspect emphasizes:
1. The presence of four linear PI controllers for each fuzzy controller structure, blended bymeans of the PI-FC operating mechanism.
2. The following design recommendations expressed in terms of relations between the
design parameters specific to linear PI controllers:
in case ofd1 andd2 type disturbance inputs:b14b2; b3ob4, (36)
in case ofd3 type disturbance inputs:b
14b
2; b
34b
4, (37)
and the parameters in (28) are calculated according to (21) with br b2.
The proposed design method is dedicated to 2-DOF TakagiSugeno PI-FCs. The method
consists of the following steps presented in unified manner for all four fuzzy controllers:
Step 1: Identify the controlled process and express the simplified mathematical model interms of the transfer function P(s) in (1), specific to integral servo systems.
Step 2: Set the values of the design parametersa,b1,b2,b3andb4of the linear continuous-time 2-DOF PI controller types, 2-DOF PI-C-r and 2-DOF PI-C-d, as function of the
desired/imposed CS performance indices taking into account the recommendations (36)and (37) and the results derived from (12)(20) assisted by Tables 14.
Step 3: Tune the parameters of the linear continuous-time 2-DOF PI controller types,2-DOF PI-C-r and 2-DOF PI-C-d, in terms of (21) for br b1and br b2and (22) forbdb3 and bdb4. Step 4: Set an adequate value of the sampling period, h, accepted by quasi-continuousdigital control and account for the presence of the zero-order hold.
Step 5: Discretize the dynamics components in the linear 2-DOF PI controller structuresand calculate the parameters of the quasi-continuous digital components employing
(26), (28) and (31).
Step 6: Set the value of the parameter Se of the 2-DOF TakagiSugeno PI-FCs inaccordance with the experience of the control systems designer and apply (38) (for the 2-
DOF PI-FCs inFig. 2(a)(c)) or (39) (for the 2-DOF PI-FC inFig. 2(d)) corresponding
to the modal equivalence principle[14]to obtain the value ofSDe:
SDeSe maxbr2fb1 ;b2gbd2fb3 ;b4g
fKrIj=jKrPjj; KdIj=jKdPjjg; j1; 4, (38)
SDeSe maxbr2fb1 ;b2gbd2fb3 ;b4g
fKrI4j=jKrP4jj; KdI4j=jKdP4jjg; j1; 4. (39)
Step 7: Set the values of the other two 2-DOF TakagiSugeno PI-FC parameters, SDrandSs, in terms of (40) that accepts the constant rmaxstep modification ofr and the 2%
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 13
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
14/24
settling time, and makes the difference between dynamic regimes concerning the
modifications of the set-point and disturbance inputs:
SDr0:02rmax; Ss1. (40)
The first three steps in the design method represent the design method for the
continuous-time linear 2-DOF PI controllers. The first five steps correspond to the design
method for the discrete-time linear 2-DOF PI controllers. Eqs. (38)(40) ensure the
approximate equivalence, accepted in Section 1, between the suggested generic structures
of 2-DOF TakagiSugeno PI-FCs and the linear 2-DOF PI controllers.
The stability analysis theorem to be presented as follows will offer valuable information
to set the values of the parameter Se in step 6 of the design method. A stable CS will be
obtained. Accepting the quasi-continuous digital control case the mathematical model of
the controlled process (1) can be transformed generally to the state-space form
_xfx; t bx; tu,xt0 x0, (41)
wherex x1 x2 . . . xnT 2D is the state vector, nAN*, _x _x1 _x2 . . . _xnT isthe derivative ofx with respect to the independent time variable t, f; b :D 0; 1 !Rnare continuous functions in t, fx; t f1x; t f2x; t . . . fnx; t T, bx; t b1x; t b2x; t . . . bnx; t T, T stands for matrix transposition, and the disturbanceis absent. The generality of the problem is not reduced because all controllers have integral
components that deal with the rejection of constant load-type disturbances. The particular
expressions of the variables and functions in (41) become (42) for the given processcharacterized by n2:
x x1 x2T; x1ery; x2 _x1; fx; t x2 x2=TSt; bx; t 0 kP=TST,(42)
where the time-variable character has been introduced to increase the generality when
parametric disturbances can occur making the process belong the class of time-variant
systems.
The rule base in the fuzzy controllers consists ofrBfuzzy control rules. The ith rule is
Rule i :IF x1 IS Xi;1 AND . . . AND xn IS Xi;n THEN uuix; i1; rB; rB2,(43)
where Xi;1;. . .; Xi;n are the fuzzy sets that describe the linguistics terms of the inputvariables,ui(x) is the control signal produced by the ith rule and the function AND is the
MAX operator. ui can be a constant (that is also the case of Mamdani fuzzy controllers
with singleton consequents) or a function depending on the state vector. For the given
inference engine each fuzzy rule generates a certain firing strength
aix
minm X
i;1 x1
;m X
i;2x2
;. . .;m X
i;n xn
; 0
ai
x
1; i
1; rB, (44)
with the assumption
8x2D9aia0; i1; rB. (45)
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]14
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
15/24
For the given defuzzification method the expression of the control signal is
u XrB
i
1
aiui
!,X
rB
i
1
ai
!. (46)
The stability analysis theorem to be presented as follows is based on Lyapunovs theorem
for time-varying systems [33,34]. The theorem ensures sufficient uniform asymptotic
stability conditions for the fuzzy control systems. A Lyapunov function candidate
V :D 0; 1 !R; Vx; t gtxTPx (47)is suggested, where PARn n is a constant positive definite matrix and g:[0,N)-[0,N) is a
continuously differentiable function. The derivative of V with respect to time with the
system being subject to the trajectory (41) is
_Vx; t _gtxTPxgt _xTPxxTP _x _gtxTPxgtfx; tbx; tuTPxgtxTPfx; t bx; tu Fx; t Bx; tu, (48)
with the following notations:
Fx; t gtfx; tTPxgtxTPfx; t _gtxTPx,Bx; t gtbx; tTPxgtxTPbx; t. (49)
The derivative ofVwith respect to time with the system accepted on the trajectory (41)
and calculated for u
uk(x) is
_Vkx; t Fx; t Bx; tukx. (50)Use is made of (50) and the stability analysis theorem is expressed in terms of Theorem 1.
Theorem 1. Let the fuzzy control systems be characterized by one of the four 2-DOF
Takagi Sugeno PI-FCs and the process(41). Let x0ADCRn be an equilibrium point for(41) and V the Lyapunov function candidate (47) such that the conditions (51) and (52) are
fulfilled:
Vx; t
W1
x, (51)
_Vkx; t W2kx, (52)for8k1; rB; 8t0; 8x2D,where W1 and Wk2 are continuous positive definite functionson D. Then x0 will be uniformly asymptotically stable.
Proof. From the hypotheses of Theorem 1,8t0; 8x2D, (50) and (52) result inFx; t Bx; tukx W2kx; k1; rB. (53)
Next (53) is multiplied by ak(x) and the sum is calculated:
Fx; tXrBk1
akx Bx; tXrBk1
akxukx XrBk1
W2kxakx. (54)
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 15
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
16/24
The division byPrB
k1akx leads to
Fx; t Bx; tX
rB
k
1
akxukx" #,
XrB
k
1
akx" #
X
rB
k
1
W2kxakx" #,
XrB
k
1
akx" #
.
(55)
Use is made of (48) and the final result is
_Vx; t XrBk1
W2kxakx" #,XrB
k1akx
" #. (56)
Summarizing, the conditions (51) and (56) fulfil Lyapunovs theorem for time-varying
systems. Therefore, the equilibrium point at the origin will be uniformly asymptotically
stable and the proof is complete. Concluding, Theorem 1 ensures sufficient stability
conditions for the accepted fuzzy control systems and it can be employed in setting the
values ofSe.
4. Experimental results
In order to validate the proposed design methods, we implemented 2-DOF linear PI
controllers and TakagiSugeno PI-FCs for the control of a laboratory DC drive (AMIRA
DR300).
The experimental setup is illustrated in Fig. 4. The DC motor is loaded by a current
controlled DC generator mounted on the same shaft, and the drive has built-in analog
current controllers for both DC machines having rated speed equal to 3000 rpm, ratedpower equal to 30 W, and rated current equal to 2 A. An A/D D/A data converter card is
used as an interface between the digital speed-controller and the DC motor. The speed
sensors are a tacho generator and an additional incremental rotary encoder mounted
on the drive-shaft. The controlled process can be well approximated by the transfer
function P(s) in (1), decomposed according to (2) with kP4900, kP11, kP24900,TS0.035 s.
ARTICLE IN PRESS
Fig. 4. Experimental setup.
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]16
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
17/24
The design steps presented in Section 3 were applied for the PI-FCs and the main
parameters involved in the design of 2-DOF linear PI controllers and TakagiSugeno
PI-FCs are:
the design parameters: a1, b19, b26, b316, b44, the parameters of the continuous-time linear controllers: kC10.0019, Ti10.315 s,kC20.0024, Ti,20.24 s,kC30.0015, Ti30.56 s,kC40.0029, Ti40.14 s, the parameters of the digital PI controllers:h0.005 s, in case of set-point filter 2-DOFPI-FC and feedforward 2-DOF PI-FC: KrP10:0018, KrI10:0062 (corresponding tokC1andTi1),K
rP20:0021,KrI20:0113 (corresponding tokC2andTi2),Kd3P10:0014,
Kd3I1 0:0026 (corresponding tokC3andTi3), Kd3P20:0024,Kd3I2 0:0208 (correspond-ing tokC4andTi4), in case of feedback 2-DOF PI-FC: K
rP1 1:5104,KrI10:0062,
KrP2
2:8
104,KrI2
0:0113 , Kd3P1
6:5
104,Kd3I1
0:0026,Kd3P2
5:2
104,
Kd3I2 0:0208 , in case of component-separated 2-DOF PI-FC: KrP10:0794, KrI13:1746 (corresponding to Ti1), K
rP20:119 , KrI24:7619 (corresponding to Ti2),
Kd3P10:0446, Kd3I1 1:7857 (corresponding to Ti3), Kd3P20:1786, Kd3I2 7:1429
ARTICLE IN PRESS
Fig. 5. Speed response of conventional CS with 2-DOF PI controllers without load (a), forr2500 rpm and 5 speriod of 10% d3 rated load (b).
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 17
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
18/24
(corresponding to Ti4), and the elements without dynamics together with the set-point
filter implemented for bb2, the parameters of B-FC as part of all four fuzzy controllers: Se750,SAr50, Ss1,
the parameterSAeof B-FC:SAe
8695.7 in case of set-point filter 2-DOF PI-FC and of
feedforward 2-DOF PI-FC, SAe 30000 in case of feedback 2-DOF PI-FC and ofcomponent-separated 2-DOF PI-FC.
The four 2-DOF TakagiSugeno PI-fuzzy controllers designed here were tested by real-
time experiments and compared with the conventional 2-DOF controllers designed for
bb2. The speed responses exhibited by the designed linear and fuzzy control systems arepresented in Figs. 59 for low speed patterns. The results are presented with respect to
the modifications of the set-pointrand thed3type load disturbance input (according to the
definition inFig. 1(e)).
The experimental results prove that all fuzzy controllers outperform the linear 2-DOF PI
controllers accounting for the fact that the behaviour of all four CSs with linear 2-DOF PI
ARTICLE IN PRESS
Fig. 6. Speed response of CS with set-point filter 2-DOF PI-FC without load (a), forr2500 rpm and 5 s periodof 10% d3 rated load (b).
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]18
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
19/24
controllers is very close, the control schemes being equivalent as mentioned in Section 1.
The comparison of experimental results proves that the smallest settling time and
overshoot are exhibited by
in case of set-point modifications: the CS with feedforward 2-DOF PI-FC, next the CSwith component-separated 2-DOF PI-FC, the CS with feedback 2-DOF PI-FC and the
CS with set-point filter 2-DOF PI-FC,
in case ofd3type load disturbance input modifications: the CS with feedback 2-DOF PI-FC, next the CS with component-separated 2-DOF PI-FC, the CS with set-point filter
2-DOF PI-FC and the CS with feedforward 2-DOF PI-FC.
5. Conclusions
This paper presents a new framework for the design of generic two-degree-of-freedom
(2-DOF), linear and fuzzy, controllers dedicated to a class of integral processes specific to
ARTICLE IN PRESS
Fig. 7. Speed response of CS with feedforward 2-DOF PI-FC without load (a), forr2500 rpm and 5 s period of10%d3 rated load (b).
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 19
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
20/24
servo systems. These controllers can be implemented as low-cost automation solutions
because:
their relatively simple and transparent design method requires small computational costand ensures easy tuning,
the controller structures are relatively simple.The main advantages of the FC structures with respect to the linear ones are that they
exhibit a bumpless transfer from one linear PI controller to another, and they have a very
good behaviour in both set-point tracking and regulation (for all three load disturbance
types).
The rule base defined in Section 3 has a special formulation that improves the CS
behaviour when ekand Dekhave the same sign. This situation illustrates the downshoot
specific to non-minimum phase systems with right half-plane zeros. Therefore, theapplication areas of the fuzzy controller structures proposed here can be enlarged.
The proposed design methods have been shown to be very effective in set-point tracking
and load disturbance regulation when dealing with control of real-world processes because
ARTICLE IN PRESS
Fig. 8. Speed response of CS with feedback 2-DOF PI-FC without load (a), for r2500 rpm and 5 s period of10%d3 rated load (b).
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]20
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
21/24
the industrial applications involve various integral servo systems. Experimental results
discussed in the paper validate both the design methods and the controller structures in
case of low speed patterns. The FC structures appear to be particularly appropriate forindustrial applications due to their low-cost and compatibility with linear 2-DOF PI
controllers.
The apparent complexity of the proposed, seven-step tuning procedure is in fact reduced
significantly due to (12)(20) andTables 14which connect the performance specifications
and the design parameters. They support the practitioners in the choice of the linear or
fuzzy controllers such that to fulfil the performance specifications.
The four linear control systems are identical. Their fuzzy counterparts are different but
they exhibit similar behaviours for the accepted processes. However all fuzzy control
system structures can be designed relatively easily to fulfil the performance specifications.
The choice of the appropriate fuzzy controllers for a certain process such that the desires/imposed performance indices are obtained represents the practitioners option. They
should account for the most convenient way to insert fuzzy logic into an already
implemented linear 2-DOF controller dealing with that process.
ARTICLE IN PRESS
Fig. 9. Speed response of CS with component-separated 2-DOF PI-FC without load (a), forr2500 rpm and 5speriod of 10% d3 rated load (b).
R.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]] 21
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
22/24
To improve the CS performance indices these controller structures can be extended with
minor effort to advanced fuzzy controller structures [10,20,27,45]. Further research will be
focused on fuzzy controllers extending the area of applicability to servo systems with
essential nonlinearities, but together with appropriate analyses including the stability
analysis due to many variables and parameters involved [4852].
Acknowledgements
This work was supported by the Budapest Tech Polytechnical Institution and the
Politehnica University of Timisoara in the framework of the Hungarian-Romanian
Intergovernmental Science & Technology Cooperation Program, and by the CNCSIS &
CNMP of Romania.
References
[1] K.-J. Astrom, T. Hagglund, The future of PID control, Control Eng. Pract. 9 (11) (2001) 11631175.
[2] K.-J. Astrom, T. Hagglund, PID Controllers Theory: Design and Tuning, Instrument Society of America,
Research Triangle Park, NC, 1995.
[3] G. Prashanti, M. Chidambaram, Set-point weighted PID controllers for unstable systems, J. Franklin Inst.
337 (23) (2000) 201215.
[4] M. Araki, H. Taguchi, Two-degree-of-freedom PID controllers, Int. J. Control Automat. Syst. 1 (4) (2003)
401411.
[5] A. Visioli, A new design for a PID plus feedforward controller, J. Process Contr. 14 (4) (2004) 457463.
[6] P. Albertos, Fuzzy logic control: light and shadow, IFAC Newsletter 26 (3) (2002) 12.
[7] M. Oosterom, R. Babuska, Design of a gain-scheduling mechanism for flight control laws by fuzzy clustering,
Control Eng. Pract. 14 (7) (2006) 769781.
[8] M. Sunar, O. Toker, Substructural control of fuzzy nonlinear flexible structures, J. Franklin Inst. 344 (5)
(2007) 646657.
[9] A. Bagis, D. Karaboga, Evolutionary algorithm-based fuzzy PD control of spillway gates of dams,
J. Franklin Inst. 344 (8) (2007) 10391055.
[10] K. Michels, F. Klawonn, R. Kruse, A. Nurnberger, Fuzzy Control: Fundamentals, Stability and Design of
Fuzzy Controllers, Springer-Verlag, Berlin, Heidelberg, New York, 2006.
[11] L. Wen, Y. Hori, Vibration suppression using single neuron-based PI fuzzy controller and fractional-order
disturbance observer, IEEE Trans. Ind. Electr. 54 (1) (2007) 117126.
[12] B.M. Mohan, A. Sinha, Analytical structures for fuzzy PID controllers?, IEEE Trans. Fuzzy Syst. 16 (1)
(2008) 5260.
[13] R. Shahnazi, H.M. Shanechi, N. Pariz, Position control of induction and DC servomotors: a novel adaptivefuzzy PI sliding mode control, IEEE Trans. Energy Conv. 23 (1) (2008) 138147.
[14] R.-E. Precup, S. Preitl, I.J. Rudas, M.L. Tomescu, J.K. Tar, Design and experiments for a class of fuzzy
controlled servo systems, IEEE/ASME Trans. Mechatronics 13 (1) (2008) 2235.
[15] S. Galichet, L. Foulloy, Fuzzy controllers: synthesis and equivalences, IEEE Trans. Fuzzy Syst. 3 (2) (1995)
140148.
[16] J. Jantzen, Foundations of Fuzzy Control, John Wiley & Sons, Chichester, 2007.
[17] H. Ying, Theory and application of a novel Takagi-Sugeno fuzzy PID controller, Inform. Sci. 123 (34)
(2000) 281293.
[18] G.K.I. Mann, R.G. Gosine, Three-dimensional min-max-gravity based fuzzy PID inference analysis and
tuning, Fuzzy Sets Syst. 156 (2) (2005) 300323.
[19] T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE
Trans. Syst. Man Cyb. 15 (1) (1985) 116132.[20] A. Sala, T.M. Guerra, R. Babuska, Perspectives of fuzzy systems and control, Fuzzy Sets Syst. 156 (3) (2005)
432444.
[21] C.M. Liaw, S.Y. Cheng, Fuzzy two-degrees-of-freedom speed controller for motor drives, IEEE Trans. Ind.
Electr. 42 (2) (1995) 209216.
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]22
Please cite this article as: R.-E. Precup, et al., Generic two-degree-of-freedom linear and fuzzy controllers for
integral processes, J. Franklin Inst. (2009), doi:10.1016/j.jfranklin.2009.03.006
http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.jfranklin.2009.03.006 -
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
23/24
-
8/13/2019 Generic Two Degree of Freedomlinearandfuzzy
24/24
[50] C. Prieur, R. Goebel, A.R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems,
IEEE Trans. Automat. Control 52 (11) (2007) 21032117.
[51] R. Dargahi, A.H.D. Markazi, HN
-optimal digital redesign method, J. Franklin Inst. 344 (5) (2007) 553564.
[52] J.S.-H. Tsai, C.-L. Wei, S.-M. Guo, L.S. Shieh, C.R. Liu, EP-based adaptive tracker with observer and fault
estimator for nonlinear time-varying sampled-data systems against actuator failures, J. Franklin Inst. 345 (5)
(2008) 508535.
ARTICLE IN PRESSR.-E. Precup et al. / Journal of the Franklin Institute ] (]]]]) ]]]]]]24
top related